Invertibility of Bass-Connell-Wright polynomial maps (Q1318053)

Let \(I : \mathbb{C}^ q \to \mathbb{C}^ q\) be the identity, \(H : \mathbb{C}^ q \to \mathbb{C}^ q\) a homogeneous polynomial mapping of degree greater than 1 with the everywhere nilpotent Jacobian matrix \(H'\) and \(k\) the least integer for which \([H'(x)]^ k\) is identically zero. -- The following theorem is proved: Let \(F = I - H\): \(\mathbb{C}^ q \to \mathbb{C}^ q\) be a polynomial mapping where \([H'(x)]^ q \equiv 0\) and \(k \geq 3\). If \(\bigcup \{\ker [H'(x)]^{k-1}\): \(H(x) = 0\), \(x \neq 0\} \neq \mathbb{C}^ q\), then \(F\) is invertible. It is a pity that the author does not give any example of a mapping fulfilling the assumptions of the theorem. The reviewer does not know any such example, either.